Conditions on one forms I am trying to solve exercise 8.3 from Lightman's problem book, but I don't know where to start to get a sufficient and necessary condition on a field of one forms $\tilde\sigma$ for there to exist a function f such that $\tilde\sigma $ = $\tilde{df}$.
I even tried to understand the solution, but I didn't. Can you help me, please?
 A: Here is my attempt. For clarity, the question refers to 3-dimensional Euclidean space.
Let $f$ be a scalar function that is at least two times continuously differentiable. It is important to note ${\bf \tilde\sigma}$ is a $1$-form field.
The scalar function $f$ is said to be a $0$-form. If we can write
$$
{\bf \tilde \sigma} = \tilde{df} = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dy\,.
$$
we can say ${\bf \tilde\sigma}$ is exact.
By the Poincare Lemma we can say that exterior derivative of ${\bf \tilde\sigma}$ vanishes, i.e.,
$$
d(\tilde{df}) = 0\,.
$$
But the exterior derivative of $\tilde \sigma$ is
$$
d\tilde\sigma  = \left(\frac{\partial^2f}{\partial x\partial y} - \frac{\partial^2f}{\partial y\partial x}\right)dx \wedge dy +\left(\frac{\partial^2f}{\partial x\partial z} - \frac{\partial^2f}{\partial z\partial x}\right)dx \wedge dz + \left(\frac{\partial^2f}{\partial y\partial z} - \frac{\partial^2f}{\partial z\partial y}\right)dy \wedge dz\,.
$$
Since $d\tilde\sigma = 0$ it must be the case that
$$
\frac{\partial^2f}{\partial x\partial y} = \frac{\partial^2f}{\partial y\partial x}\,,\quad 
\frac{\partial^2f}{\partial x\partial z} = \frac{\partial^2f}{\partial z\partial x}\,,\quad 
\frac{\partial^2f}{\partial y\partial z} = \frac{\partial^2f}{\partial z\partial y}\,.
$$
This is exactly the condition that is meant by $\sigma_{i,j} = \sigma_{j,i}\,$ as Lightman et al. suggest in their solution because
$$
\tilde\sigma_{i} = \frac{\partial f}{\partial x^{i}}\,,
$$
where $i=1,2,3$ and $x^1=x, x^2 = y, x^3=z\,.$
I didn't use the content at the link https://en.wikipedia.org/wiki/Differential_form but I think exact differential forms are mentioned there. I checked "Tensors, Differential Forms and Variational Principles" by David Lovelock and Hanno Rund.
